Question: Simplify the following expression: $ z = \dfrac{5}{3} - \dfrac{a + 1}{-2a + 6} $
Answer: In order to subtract expressions, they must have a common denominator. Multiply the first expression by $\dfrac{-2a + 6}{-2a + 6}$ $ \dfrac{5}{3} \times \dfrac{-2a + 6}{-2a + 6} = \dfrac{-10a + 30}{-6a + 18} $ Multiply the second expression by $\dfrac{3}{3}$ $ \dfrac{a + 1}{-2a + 6} \times \dfrac{3}{3} = \dfrac{3a + 3}{-6a + 18} $ Therefore $ z = \dfrac{-10a + 30}{-6a + 18} - \dfrac{3a + 3}{-6a + 18} $ Now the expressions have the same denominator we can simply subtract the numerators: $z = \dfrac{-10a + 30 - (3a + 3) }{-6a + 18} $ Distribute the negative sign: $z = \dfrac{-10a + 30 - 3a - 3}{-6a + 18}$ $z = \dfrac{-13a + 27}{-6a + 18}$ Simplify the expression by dividing the numerator and denominator by -1: $z = \dfrac{13a - 27}{6a - 18}$